If the maximum tension the ring can withstand is Fmax and its linear mass density is d. The maximum permissible linear velocity of a rotating thin lead ring (axis of rotation is the axis of the ring) is

Engineering

Physics

Center of Mass

Circular Motion in Vertical Plane

Forces and Types of Forces

Question

If the maximum tension the ring can withstand is F_max and its linear mass density is d. The maximum permissible linear velocity of a rotating thin lead ring (axis of rotation is the axis of the ring) is

$\sqrt{\frac{F_{Max}}{d}}$

$\sqrt{\frac{F_{Max}}{3 d}}$

$\sqrt{\frac{F_{Max}}{2 d}}$

$\sqrt{\frac{2 F_{Max}}{d}}$

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For a thin ring rotating about its axis, the centripetal force is provided by tension. Consider a small element of mass dm. The centripetal force on this element is dm × v²/R. This force is balanced by the tension forces from both sides. For a small angle dθ, the net inward force from tension is 2T sin(dθ/2) ≈ T dθ. Equating forces: T dθ = (d × R dθ) × v²/R, where d is linear mass density. Simplifying: T = d v². Thus maximum velocity occurs when T = F_max, giving v_max = √(F_max/d).

Final answer: $\sqrt{\frac{F_{M a x}}{d}}$